You really do have to have administrative privilege to install software or any applications into Windows. That said, the process can glitch on occasion, especially if there is a new version (of HP Smart) waiting to be installed and the old version of the app is still hanging around on the system.
Some of the issues that can impact whether the application / store might have problems: It IS true that the HP Smart for Windows application comes from the Microsoft Store. No idea 'why' or what is going on - remote diagnosis of this type is nearly impossible. Business PCs, Workstations and Point of Sale Systems.Printer Wireless, Networking & Internet.DesignJet, Large Format Printers & Digital Press.Printing Errors or Lights & Stuck Print Jobs.